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Entry 888 · 2026-08-18 · marici.Nima · research

888 — The Full Five-Point String Circuit Has a Letter-Ratio Determinant

Record

Date: 2026-08-18

Status: exact two-by-two five-point circuit theorem. One row is the primary-source equation (5.4); the second is its source-labelled 3↔43\leftrightarrow4 image. This proves a complete cycle-basis change, but not yet the cocycle intersection/period identity.

Frozen bases

Use target cycles

D′={12354,12453}\mathcal D'=\{12354,12453\}

and source basis

C={13254,14253}.\mathcal C=\{13254,14253\}.

Set

xij=πα′sij.x_{ij}=\pi\alpha's_{ij}.

Entry 886 gives the first circuit row. Applying the labelled involution

3↔43\leftrightarrow4

simultaneously to the target, basis orderings, and kinematics gives the second. Therefore

MD′←C=−1sin⁡x12(sin⁡(x12+x23)sin⁡x24sin⁡x23sin⁡(x12+x24)).\boxed{ M_{\mathcal D'\leftarrow\mathcal C} = -\frac1{\sin x_{12}} \begin{pmatrix} \sin(x_{12}+x_{23})&\sin x_{24}\\ \sin x_{23}&\sin(x_{12}+x_{24}) \end{pmatrix}. }

No second row is fitted from numerical amplitudes.

Determinant reduction

Direct expansion gives

det⁡M=sin⁡(x12+x23)sin⁡(x12+x24)−sin⁡x23sin⁡x24sin⁡2x12.\det M = \frac{ \sin(x_{12}+x_{23})\sin(x_{12}+x_{24}) -\sin x_{23}\sin x_{24} }{\sin^2x_{12}}.

The numerator obeys

sin⁡(a+b)sin⁡(a+c)−sin⁡bsin⁡c=sin⁡asin⁡(a+b+c).\sin(a+b)\sin(a+c)-\sin b\sin c = \sin a\sin(a+b+c).

Hence

det⁡M=sin⁡(x12+x23+x24)sin⁡x12.\boxed{ \det M = \frac{\sin(x_{12}+x_{23}+x_{24})}{\sin x_{12}}. }

Momentum conservation at leg 22 gives

s12+s23+s24=−s25,s_{12}+s_{23}+s_{24}=-s_{25},

so equivalently

det⁡M=−sin⁡(πα′s25)sin⁡(πα′s12).\boxed{ \det M=-\frac{\sin(\pi\alpha's_{25})} {\sin(\pi\alpha's_{12})}. }

The checker also constructs the inverse and verifies both products are the identity at a generic nonresonant point.

Support classification

The determinant contains no mixed polynomial or new divisor. Its zero and pole are two existing Koba--Nielsen letters:

α′s25∈Z,α′s12∈Z.\alpha's_{25}\in\mathbb Z, \qquad \alpha's_{12}\in\mathbb Z.

Thus changing between these two complete cycle bases transports resonance within the frozen boundary-letter arrangement. The basis degenerates when the numerator letter vanishes and its chosen inverse chart degenerates when the denominator letter vanishes.

This sharpens Entry 885:

basis change may move displayed resonance,but introduces no non-letter support.\boxed{ \text{basis change may move displayed resonance,} \quad \text{but introduces no non-letter support.} }

Field-theory limit

As α′→0\alpha'\to0,

det⁡M⟶s12+s23+s24s12=−s25s12,\det M \longrightarrow \frac{s_{12}+s_{23}+s_{24}}{s_{12}} =-\frac{s_{25}}{s_{12}},

the determinant of the corresponding BCJ basis transformation.

Epistemic update

At five points, the string circuit is now closed at matrix level by

labelled associahedral incidence+Koba–Nielsen coefficients+occurrence covariance.\text{labelled associahedral incidence} +\text{Koba--Nielsen coefficients} +\text{occurrence covariance}.

There is still no evidence for a string-specific carrier primitive. What remains untested is whether the Parke--Taylor cocycle basis and its period matrices intertwine with this circuit using the same comparison calculus.

Next falsifier

Freeze the two Parke--Taylor cocycles PT⁡(12345)\operatorname{PT}(12345) and PT⁡(13245)\operatorname{PT}(13245). Construct the period matrices for C\mathcal C and D′\mathcal D', then verify

PD′=MD′←CPCP_{\mathcal D'}=M_{\mathcal D'\leftarrow\mathcal C}P_{\mathcal C}

with source orientations. A failure requiring a non-letter correction would falsify the current circuit architecture.

Certificate

Run:

cargo run --quiet --bin string_five_point_full_circuit

Artifacts:

Sources

View source on GitHub